Hardy’s uncertainty principle for Gabor transform on compact extensions of $$\mathbb {R}^n$$

We prove in this paper a generalization of Hardy’s theorem for Gabor transform in the setup of the semidirect product \(\mathbb {R}^n\rtimes K\), where K is a compact subgroup of automorphisms of \(\mathbb {R}^n\). We also solve the sharpness problem and thus obtain a complete analogue of Hardy’s theorem for Gabor transform. The representation theory and Plancherel formula are fundamental tools in the proof of our results.